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  <div class="section" id="numpy-fft-fft">
<h1>numpy.fft.fft<a class="headerlink" href="#numpy-fft-fft" title="Permalink to this headline">¶</a></h1>
<dl class="function">
<dt id="numpy.fft.fft">
<code class="sig-prename descclassname">numpy.fft.</code><code class="sig-name descname">fft</code><span class="sig-paren">(</span><em class="sig-param">a</em>, <em class="sig-param">n=None</em>, <em class="sig-param">axis=-1</em>, <em class="sig-param">norm=None</em><span class="sig-paren">)</span><a class="reference external" href="https://github.com/numpy/numpy/blob/v1.18.1/numpy/fft/_pocketfft.py#L98-L189"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#numpy.fft.fft" title="Permalink to this definition">¶</a></dt>
<dd><p>Compute the one-dimensional discrete Fourier Transform.</p>
<p>This function computes the one-dimensional <em>n</em>-point discrete Fourier
Transform (DFT) with the efficient Fast Fourier Transform (FFT)
algorithm [CT].</p>
<dl class="field-list">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><dl>
<dt><strong>a</strong><span class="classifier">array_like</span></dt><dd><p>Input array, can be complex.</p>
</dd>
<dt><strong>n</strong><span class="classifier">int, optional</span></dt><dd><p>Length of the transformed axis of the output.
If <em class="xref py py-obj">n</em> is smaller than the length of the input, the input is cropped.
If it is larger, the input is padded with zeros.  If <em class="xref py py-obj">n</em> is not given,
the length of the input along the axis specified by <em class="xref py py-obj">axis</em> is used.</p>
</dd>
<dt><strong>axis</strong><span class="classifier">int, optional</span></dt><dd><p>Axis over which to compute the FFT.  If not given, the last axis is
used.</p>
</dd>
<dt><strong>norm</strong><span class="classifier">{None, “ortho”}, optional</span></dt><dd><div class="versionadded">
<p><span class="versionmodified added">New in version 1.10.0.</span></p>
</div>
<p>Normalization mode (see <a class="reference internal" href="../routines.fft.html#module-numpy.fft" title="numpy.fft"><code class="xref py py-obj docutils literal notranslate"><span class="pre">numpy.fft</span></code></a>). Default is None.</p>
</dd>
</dl>
</dd>
<dt class="field-even">Returns</dt>
<dd class="field-even"><dl class="simple">
<dt><strong>out</strong><span class="classifier">complex ndarray</span></dt><dd><p>The truncated or zero-padded input, transformed along the axis
indicated by <em class="xref py py-obj">axis</em>, or the last one if <em class="xref py py-obj">axis</em> is not specified.</p>
</dd>
</dl>
</dd>
<dt class="field-odd">Raises</dt>
<dd class="field-odd"><dl class="simple">
<dt><strong>IndexError</strong></dt><dd><p>if <em class="xref py py-obj">axes</em> is larger than the last axis of <em class="xref py py-obj">a</em>.</p>
</dd>
</dl>
</dd>
</dl>
<div class="admonition seealso">
<p class="admonition-title">See also</p>
<dl class="simple">
<dt><a class="reference internal" href="../routines.fft.html#module-numpy.fft" title="numpy.fft"><code class="xref py py-obj docutils literal notranslate"><span class="pre">numpy.fft</span></code></a></dt><dd><p>for definition of the DFT and conventions used.</p>
</dd>
<dt><a class="reference internal" href="numpy.fft.ifft.html#numpy.fft.ifft" title="numpy.fft.ifft"><code class="xref py py-obj docutils literal notranslate"><span class="pre">ifft</span></code></a></dt><dd><p>The inverse of <a class="reference internal" href="#numpy.fft.fft" title="numpy.fft.fft"><code class="xref py py-obj docutils literal notranslate"><span class="pre">fft</span></code></a>.</p>
</dd>
<dt><a class="reference internal" href="numpy.fft.fft2.html#numpy.fft.fft2" title="numpy.fft.fft2"><code class="xref py py-obj docutils literal notranslate"><span class="pre">fft2</span></code></a></dt><dd><p>The two-dimensional FFT.</p>
</dd>
<dt><a class="reference internal" href="numpy.fft.fftn.html#numpy.fft.fftn" title="numpy.fft.fftn"><code class="xref py py-obj docutils literal notranslate"><span class="pre">fftn</span></code></a></dt><dd><p>The <em>n</em>-dimensional FFT.</p>
</dd>
<dt><a class="reference internal" href="numpy.fft.rfftn.html#numpy.fft.rfftn" title="numpy.fft.rfftn"><code class="xref py py-obj docutils literal notranslate"><span class="pre">rfftn</span></code></a></dt><dd><p>The <em>n</em>-dimensional FFT of real input.</p>
</dd>
<dt><a class="reference internal" href="numpy.fft.fftfreq.html#numpy.fft.fftfreq" title="numpy.fft.fftfreq"><code class="xref py py-obj docutils literal notranslate"><span class="pre">fftfreq</span></code></a></dt><dd><p>Frequency bins for given FFT parameters.</p>
</dd>
</dl>
</div>
<p class="rubric">Notes</p>
<p>FFT (Fast Fourier Transform) refers to a way the discrete Fourier
Transform (DFT) can be calculated efficiently, by using symmetries in the
calculated terms.  The symmetry is highest when <em class="xref py py-obj">n</em> is a power of 2, and
the transform is therefore most efficient for these sizes.</p>
<p>The DFT is defined, with the conventions used in this implementation, in
the documentation for the <a class="reference internal" href="../routines.fft.html#module-numpy.fft" title="numpy.fft"><code class="xref py py-obj docutils literal notranslate"><span class="pre">numpy.fft</span></code></a> module.</p>
<p class="rubric">References</p>
<dl class="citation">
<dt class="label" id="r8913bc1f0021-ct"><span class="brackets">CT</span></dt>
<dd><p>Cooley, James W., and John W. Tukey, 1965, “An algorithm for the
machine calculation of complex Fourier series,” <em>Math. Comput.</em>
19: 297-301.</p>
</dd>
</dl>
<p class="rubric">Examples</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">np</span><span class="o">.</span><span class="n">fft</span><span class="o">.</span><span class="n">fft</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="mi">2</span><span class="n">j</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">pi</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">8</span><span class="p">)</span> <span class="o">/</span> <span class="mi">8</span><span class="p">))</span>
<span class="go">array([-2.33486982e-16+1.14423775e-17j,  8.00000000e+00-1.25557246e-15j,</span>
<span class="go">        2.33486982e-16+2.33486982e-16j,  0.00000000e+00+1.22464680e-16j,</span>
<span class="go">       -1.14423775e-17+2.33486982e-16j,  0.00000000e+00+5.20784380e-16j,</span>
<span class="go">        1.14423775e-17+1.14423775e-17j,  0.00000000e+00+1.22464680e-16j])</span>
</pre></div>
</div>
<p>In this example, real input has an FFT which is Hermitian, i.e., symmetric
in the real part and anti-symmetric in the imaginary part, as described in
the <a class="reference internal" href="../routines.fft.html#module-numpy.fft" title="numpy.fft"><code class="xref py py-obj docutils literal notranslate"><span class="pre">numpy.fft</span></code></a> documentation:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">t</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">256</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">sp</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">fft</span><span class="o">.</span><span class="n">fft</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">t</span><span class="p">))</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">freq</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">fft</span><span class="o">.</span><span class="n">fftfreq</span><span class="p">(</span><span class="n">t</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">freq</span><span class="p">,</span> <span class="n">sp</span><span class="o">.</span><span class="n">real</span><span class="p">,</span> <span class="n">freq</span><span class="p">,</span> <span class="n">sp</span><span class="o">.</span><span class="n">imag</span><span class="p">)</span>
<span class="go">[&lt;matplotlib.lines.Line2D object at 0x...&gt;, &lt;matplotlib.lines.Line2D object at 0x...&gt;]</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
</div>
<div class="figure align-default">
<img alt="../../_images/numpy-fft-fft-1.png" src="../../_images/numpy-fft-fft-1.png" />
</div>
</dd></dl>

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